Let $$p(n):=\frac{\text{number of hamiltonian graphs with $n$ nodes}}{\text{number of graphs with $n$ nodes}}$$
Since $883156024$ of the $1018997864$ graphs with $11$ nodes are hamiltonian, we have $p(11) = 0.8667$. Is it known whether $$\lim_{n\to\infty} p(n) = 1$$
and if yes, is it known how fast this convergence is ?
The following stronger result is known. Assume that the edges of the graph $G$ with $n$ vertices are drawn in mutually independently with probability $\frac{c\ln n}{n}$. Then, for a sufficiently large $c$, the probability that $G$ contains a Hamiltonian circuit tends to $1$ as $n \to \infty$. This is Theorem 2 from this old paper by Pósa, available online for free.